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V. Dynamical Diffraction Theory
Dynamical Diffraction Theory
Fraunhofer Approximation
∫⋅−ρ= rQi
ikR
00rad e)r(dVR
ePEr)Q,R(E
vvvv
λ
Θπ=
sin4Qr
k'kQvvr
−=k
k’Q
2Θ
Repetition: Kinematical Theory
S(Q): Structure factor G(Q): Lattice factor
)Q(G)Q(SPR
rE
ee)Q(fR
rPEEE
00
Atoms
G
n,n,n
)ananan(Qi
S
k
1j
rQij
00jcrystal
321
332211j
vv
4444 34444 21444 3444 21
v vvvvvv
××=
=××== ∑ ∑∑ ++−
=
−
dVk
r
R
r-R
Observer
2Θ
k’
(5.1)
(5.2)
Bragg condition: 2dhkl sinθ = λ or Q = G
Introduction: Limitations of Kinematical Theory
When (why at all?) do we need a different and better theory
What simplifications have been made in the framework of kinematical theory?
k
k’No Refraction
No Weakening
Single Scattering Event
Dynamical Diffraction Theory
k
k’No Refraction
No Weakening
Single Scattering Event
• Refraction of the X-ray waves at the surface (or at other interfaces)
• Weakening through
� Absorption� Extinction (if Bragg condition is fulfilled)
• The Bragg reflected wave can be again reflected at other parallel net-planes
� Multiple Reflections
• The Bragg reflected wave interferes with the incoming wave � Creation of a “Wave Field”
Introduction: Limitations of Kinematical Theory
Dynamical Diffraction Theory
Introduction: History
• Loss of intensity of incoming wave � Gain of intensity of diffracted wave
• Gain of intensity of incoming wave � Loss of intensity of diffracted wave
� “Dynamical Process“
• Description through new theory (Darwin 1914, Ewald 1917)
� Modification of kinematical theory is not sufficient.� Solution of Maxwell’s equation inside and outside crystal � Boundary conditions for E, D and k at the interface
Charles Galton Darwin (1887 – 1962) Paul Peter Ewald (1888 – 1985)
Dynamical Diffraction Theory
• Dynamical theory is most relevant for perfect crystals relevant
• This is why dynamical scattering was initially not able to describe the observed intensities.
• However, kinematical theory – which was developed much later – was successful
• Nowadays, along with the presence of highly perfect single crystals - the dynamical theory is well established
Dennoch hat sie (die einfache Theorie) bei der Strukturbestimmung
an Kristallen erstaunlich gute Dienste getan aus einem
absonderlichen Grund: die Kristalle, welche die Natur darbietet,
sind in ihrem Raumgitter so fehlerhaft, dass sie – so möchte man
fast sagen – keine bessere Theorie verdienen
M. v. Laue, Röntgenstrahlinterferenzen,
Akademische Verlagsanstalt, Frankfurt/Main (1960)
Dynamical Diffraction Theory
Introduction: History
• Dynamical theory is relevant for description of diffraction on highly perfect crystals, only!
• Quartz, Calcite (“Kalkspat”): Natural appearance with high perfection
• Nowadays
� Perfect artificial crystals (Silicon, Germanium, ..) � Leibniz-Institute for Crystal Growth
� Epitaxial layers with very smooth interfaces and (near) perfect crystal structure
� Anomal Absorption (Borrmann, 1941)
� Interference double-refraction (Borrmann, 1955)
� Curvature of X-ray trajectories (Hildebrandt, 1959)
Dynamical Diffraction Theory
Introduction: History
Gerhard Borrmann
(1908 – 2006)
Zweikristalltopographie in Transmissionsgeometrie
QuelleKollimatorkristall
Probe
Fo
t op
l atte
durchgehender
Strahl
Air
Crystal
Transmitted Beam Bragg Reflected Wave
Strong Refraction Effect Interference Double Refraction
Wire
X-ray sourceCollimator Crystal
Sample
Transmitted Beam
An Example Why We Need a ‘New Theory”
Photo Plate
How does Dynamical Theory Work?
Wave Optics
• Describes interaction between X-rays and a ‘medium’
• We ask: which (plane) waves are allowed inside and outside of a medium?
• For example: Length of wave vector is different inside medium as compared to vacuum (or air)
• Description of transition from vacuum to crystal via boundary conditions
Dynamical Theory
• Same procedure than above
• Solution of Maxwell’s equations (better: wave equations) inside and outside of a crystal
• Taking into account: boundary conditions
Dynamical Diffraction Theory
Refraction
How does Dynamical Theory Work?
Dynamical Diffraction Theory
Boundary conditions at the interface
�are valid for the transmitted and the diffracted wave
�Parallel components of the wave vectors are continuous across the interface
�Length of wave vector is changing (owing to different refractive index)
�Normal components of the wave vectors are thus different at both sides of the interface
However: The solutions are more complicated than in Optics
since
Planes waves as solutions for the transmitted and reflected beam are not sufficient !
How does Dynamical Theory Work?
Dynamical Diffraction Theory
The solutions are usually more complicated than in traditional Optics
since
Planes waves as solutions for the transmitted and reflected waves are not suitable !
)trk(i0eE)t,r(E ω−⋅=
vvvvv
constant amplitude � constant intensity
This interplay cannot be described by a single plane wave !
Incoming wave supplies diffracted wave and vice versa
� We need at least two plane waves with constant amplitude (no absorption)
Dynamical Diffraction Theory
How does Dynamical Theory Work?
Incoming wave supplies diffracted wave and vice versa
How can we fulfill this requirement ?
� A spatial change of intensity can be achieved by a slight difference of the length of the wave vectors (not frequency!) of these plane waves
� This is only possible for the normal components of these plane waves (since parallel components have to be continuous across the interface)
� Two waves with slightly different wave vectors undergo a beating (“Schwebung”)
� This beating occurs for both the transmitted as well as for the diffracted beam
� It turns out that transmitted and diffracted waves show exactly opposite beating, i.e. the dominance of either transmitted or diffracted wave is oscillating
We call this modulation “Pendellösung Oscillations”
Derivation of Wave Equation
Maxwell Equations: (5.3)tD
jHrottB
Erot∂
∂+=
∂
∂−=
vvv
vv
(5.4)With the dielectric Polarization P we can write for the dielectric displacement field D PED 0
vvr+ε=
For j = 0 (no free electric current) and with the Ansatz
(5.5)ti
0
ti0
e)r(H)t,r(H
e)r(D)t,r(Dω−
ω−
=
=vvvv
vvvv
(5.6)we find
HitH
DitD
vv
vv
ω−=∂
∂
ω−=∂
∂
So let us start from the very beginning! We will derive the wave equation used in dynamical theory
Five fields (E,B,H,D,j) involved! We need a wave equation for E or D only !
Dynamical Diffraction Theory
Derivation of Wave Equation
Dynamical Diffraction Theory
and with (5.3): DitD
Hrotv
vv
ω−=∂
∂= (5.7)
Since div rot H = 0: (5.8)0Ddiv =v
(5.9)For high frequencies we can approximate:
HB 0
wvµ=
We now apply the rotation operator to Eq. (5.3)
(5.10)
Dct
DiHroti
tH
rot)PD(rotrotErotrot
2
2)6.5(
00
)3.5(
00
)6.5(
00
)3.5()4.5(
0
vv
vvvv
ω=
∂
∂µωε=µωε=
=
∂
∂µε−=−=ε
Dynamical Diffraction Theory
Derivation of Wave Equation
We use of the operator relationship DD)Ddiv(gradDrotrotvvvv
∆−=∆−=
(5.11)ProtrotDc
)PE(rotrotDrotrotD 2
2)10.5(
0
)4.5( vvvvvv−
ω−=+ε−=−=∆and end up with
DEP 00
vvvχε≈χε=Finally, we can approximate:
and (5.11) reads as 0Dc
)D(rotrotD 2
2
=ω
+χ+∆vvv
Wave equation for the dielectric displacement field D
(5.12)
Homework: Derivation of (5.13)
0Dc
)D(rotrotD 2
2
=ω
+χ+∆vvv
Wave equation for the dielectric displacement field D
(5.12)
We can derive an analogous equation for the electric field E:
0E)1(c
Edivgradc
E 2
2
2
2
=χ+ω
+⋅ω
+∆vvv
(5.13)
Dynamical Diffraction Theory
Derivation of Wave Equation
The 2nd term of this equation leads to a coupling (intermixing) of the components (direction of
polarization) of E. This coupling is very small since we can approximately write:
( ) 0Ddiv1
Ddiv
1D
divEdiv0000 1
1
0
11 ==
χ+≈
χ+=
χ+εεε
vvv
v
0E)1(c
Edivgradc
E 2
2
2
2
=χ+ω
+⋅ω
+∆vvv
0E)1(c
E 2
2
=χ+ω
+∆vv
This equation is valid for each component Ei !
Remark 1
Dynamical Diffraction Theory
Derivation of Wave Equation
)r(E)r(V̂)r(E)K( 2 vvvv=+∆
)2
cK(KgraddivV̂ 2
λ
π=
ω=χ−=with
Scattering Potential Operator
)r(r4)Re(KK)r(VV̂ e022 vv
ρπ−=χ−≈χ−==
If we use the approximation div E = 0 (see remark 1) and if we neglect absorption (use only real part of dielectric polarizability) we obtain:
Dynamical Diffraction Theory
Derivation of Wave Equation
0E)1(c
Edivgradc
E 2
2
2
2
=χ+ω
+⋅ω
+∆vvv
Remark 2
Interaction with Matter
Interaction with matter (crystal) � Dielectric Dynamic Polarizability χ(r,ω)
• scalar• complex (� imaginary part describes absorption)• depends on r (� periodic electron density)• dynamic (depends on X-ray energy)
∑γω+ω−ωε
ρ−=ωχ
j j2j
2
j
0
2e
i
g
m
e)(),(
rr
(see lecture 3)
( ) )r(K
r4)r(
m
e)r(Re e2
0e20
2 vvvρ
π−=ρ
ωε−=χ
Neglecting energy dependence (No Hönl corrections)
(5.14)
)2
cK(
λ
π=
ω=
Dynamical Diffraction Theory
Relationship between χ(r,ω) and structure factor Sg
∑∑ χ+χ=χ=χg
rgiig
rg
g
rgig e)i(e)r(
v
vvvv
v
vvv
v
g20
2
g SVm
evv
ωε=χ
See Lecture 2
(5.15)
gS v gv
: Structure factor for the reciprocal lattice vector V: Volume of unit cell
(5.16)
Interaction with Matter
Dynamical Diffraction Theory
Denotation of Physical Quantities
Quantities outside of the crystal � Capital Letters (e.g. K, D, E)
Quantities inside the crystal � Small Letters (e.g.. k, d, e)
These quantities are related to each other
(K, D, E) (k,d,e)
via requirement of continuity at the interface
n1
n2
k
k
kr
kt
k||
t
r
Example:Parallel component of the wave vector is continuous
)z,xt(kK tt gg== (5.17)
Most Simple Case: (i) Bragg condition is not fulfilled(ii) Only transmitted beam inside crystal
)r(e)i(e)r( 0g
rgiig
rg
g
rgig
vv
v
vvvv
v
vvv χ=χ≈χ+χ=χ=χ ∑∑ (5.18)
Insert this into (5.12)
{{
0dc
)1(ddc
)d)ddiv(grad(d
dc
drotrotddc
)d(rotrotd
2K
2
2
02
2
00
2
2
02
2
=ω
+χ−∆=ω
+∆−χ+∆=
=ω
+χ+∆=ω
+χ+∆
vvvvvv
vvvvvv
Solution of Wave Equation: One-Beam-Case
� One-beam-case
We can treat this case as if the X-ray wave does not ‘feel’ the presence of a periodic lattice
0Dc
)D(rotrotD 2
2
=ω
+χ+∆vvv And use small letters for
solutions inside crystal
Dynamical Diffraction Theory
Since χ0 << 1: 0
0 11
1χ+
≈χ−
and we obtain: 0dkd 20 =+∆vr
220
220 Kn)1(Kk =χ+=
211n 00 χ+≈χ+=
(5.19)
(5.20)
Refractive Index
Solution of Wave Equation: One-Beam-Case
Dynamical Diffraction Theory
This is the stationary Helmholtz equation which we know from wave optics
Refractive Index
211n 00 χ+≈χ+=
∑+−
=j j
22j
j
0
2e
i
g
m
e
γωωωε
ρωχ
)(),(
rr
0)Re( 0 <χ
β+δ−= i1n (5.21)
Material δ [10−5] β [10−7] β / δ αC
Si 0,7563 1,748 0,023 0,22°GaAs 1,441 4,087 0,028 0,31°Ge 1,444 4,200 0,029 0,31°
Materials parameter δ and β(Cu-Kα1 Radiation)
� n is a complex, scalar number
� Real part describes refraction
� Re(n) < 1
� Imaginary part describes absorption
Dynamical Diffraction Theory
0dkd 20 =+∆vr
0DKD 2 =+∆vv
Vacuum Inside Medium (Crystal)
rKi0 eD)r(D
vvvvv=
rki0
0ed)r(dvvvvv
=
Possible solutions(plane waves)
PP00(r)
KK00(r)
AA(t)
PPee=P00(t)
AA(r)
kk00
(r)
kk00
(t)
KKee == K
00(r)
α
xx
yy
Medium 2 (Substrat)
Medium 1
Multitude of possible solutions for k vector
Sphere with radius K (Vacuum) k0 (Medium)
Dispersion Surface
Dynamical Diffraction Theory
Solution of Wave Equation: One-Beam-Case
PP00(r)
KK00(r)
AA(t)
PPee=P00(t)
AA(r)
kk00
(r)
kk00
(t)
KKee == K
00(r)
α
xx
yy
Medium 2 (Substrat)
Medium 1
Dispersion Surface+
Tangential Condition Excitation Points
Dynamical Diffraction Theory
Solution of Wave Equation: One-Beam-Case
Horizontal components of the wave vectors: these are all identical for all three waves
Normal components of the wave vectors: they are different for all three waves
α=−== sinKKKK )r(y0
)t(y0ey α: angle of incidence
022
ex02
y sinKK)1(Kk χ+α=−χ+=
Vacuum
Medium
Dynamical Diffraction Theory
Solution of Wave Equation: One-Beam-Case
0sinKK)1(Kk 022
ex02
y =χ+α=−χ+=Medium (α‘=0)
( ) e0
0cr
2Re ρπ
λ=δ=χ−≈α
Critical Angle of Total External Reflection
!
Cu Kα (λ = 1.541 Å )
Material δ [10−5] β [10−7] β / δ αC
Si 0,7563 1,748 0,023 0,22°GaAs 1,441 4,087 0,028 0,31°Ge 1,444 4,200 0,029 0,31°
Dynamical Diffraction Theory
Solution of Wave Equation: One-Beam-Case
)r()t()r()t( ddDD +=+
( ) ( ))r()t()t(y
)r()t()t(y ddkDDK −=−
Horizontal components of D and H are continuous at the interface between two media
0d )r( =Media are semi infinite:
yy
yy)t(
)r(
kK
kK
D
Dr
+
−==
yy
y)t(
)t(
kK
K2
D
dt
+==
(for s-Polarization)
Fresnel reflection coefficient transmission coefficient
Comment: The Fresnel coefficients are different for p- and s-Polarization, however, these differences vanish for small angles of incidence.
(5.22)
Dynamical Diffraction Theory
Solution of Wave Equation: One-Beam-Case
2rR =2tT =
For the reflected and transmitted intensities we can thus write:
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,610-5
10-4
10-3
10-2
10-1
100
101
αcR
efle
ktiv
ität
α (Grad)
Calculation:
• Semi infinite Silicon
• λ = 1.54 Å (αc = 0.22°)
• For a more detailed discussion:See Lecture 8
Dynamical Diffraction Theory
Solution of Wave Equation: One-Beam-Case
If we fulfill Bragg condition:
� beyond the transmitted (refracted) beam also other ‘beams’ show up inside the crystal
� the (periodic) spatial dependence of the dielectric polarizability has to be taken into account
� We use the same ansatz as applied for solution of Schrödinger equation in solid state physics:
0Dc
)D(rotrotD 2
2
=ω
+χ+∆vvv
)r(v
χ=χwith
∑∑ χ+χ=χ=χg
rgiig
rg
g
rgig e)i(e)r(
v
vvvv
v
vvv
v
∑=g
rgig
rki e)k(de)r(dv
vvv
vv vvv(5.23)Bloch waves
)r(ue)r(d krki vvv
vvv
=
Bloch function, Expansion of d(r) into plane waves
(5.24)
Dynamical Diffraction Theory
Solution of Wave Equation: Two- and Multi-Beam-Case
)r(uee)k(de)r(d krki
g
rgig
rki vvvvv
vv
v
vvv
vv
== ∑
Remarks: (i) Bloch functions exhibit periodicity of the 3D lattice
(ii) Wave fields do not exhibit periodicity of the 3D lattice
(iii) Intensities exhibit periodicity of the 3D lattice
)r(dvv
2)r(dvv
)r(ukv
v
)k0kk(gkk 0g
vvvvvvvv =+=+= (5.25)Let us define:
Insertion of Bloch waves (5.23) and expansions of χ(r) (5.18) into the wave equation (5.12):
{ } 0ddk
Kk
'ggg'g'ggg02
g
22g
=⋅χ−⋅
χ−
−∑≠
−vvvvvvv
v
v vvv
v
(5.26)
{ }g'gd vvv
: Component of perpendicular to (Direction of polarization!)'gdvv
gk vv
222 cK ω=
(5.23)
Home work: Derivation of Equation (5.26)
Dynamical Diffraction Theory
Solution of Wave Equation: Two- and Multi-Beam-Case
{ } 0ddk
Kk
'ggg'g'ggg02
g
22g
=⋅χ−⋅
χ−
−∑≠
−vvvvvvv
v
v vvv
v
(5.26)
Basic equations of dynamical theory
� Infinite set of algebraic equations instead of a differential equation
� This implies the presence of an infinite number of partial plane waves inside the crystal
Dynamical Diffraction Theory
Solution of Wave Equation: Two- and Multi-Beam-Case
Let us assume now: Just two ‘strong’ waves are present inside crystal
� transmitted (refracted) and Bragg reflected wave
� with amplitudes and
� with wave vectors and
� intensity is periodic
0dv
gdvv
0kv
gkk 0gvvv
v +=
( ) ( ) rgsinddIm2rgcosddRe2IIededI *go
*gog0
2rki
grki
0go
vvvvvvvvv vvvv
+++=+= (5.27)
Pendellösung Oscillations
The periodic wave field (intensity!) inside the crystal can be written as superposition between two plane waves, which are maximal
• directly on the net-planes (here, the electron density is elevated)
and
• between adjacent net-planes (here, the electron density is reduced)
( ) ( ) rgsinddIm2rgcosddRe2IIededI *go
*gog0
2rki
grki
0go
vvvvvvvvv vvvv
+++=+= (5.27)
The two partial waves experience different electron densities (better: dielectric polarizabilities) and thus exhibit a slightly different length of the wavevector (owing to continuity requirement only in vertical direction)
‘Beating‘ of Intensities � Pendellösung oscillations
Dynamical Diffraction Theory
Pendellösung =
� interference of two plane waves
� for either the transmitted or the reflected waves
� the intensity oscillates (“pendelt”) between transmitted and reflected waves
Dynamical Diffraction Theory
Pendellösung Oscillations
Surface
Transmitted Wave Reflected Wave
( ) ( )π
ππππσσ θ≈××==
j
jiBijj]j[ii]j[i d
dd2cosdssdunddd
vvvvv
π-Polarization: Component of d within plane of incidence σ-Polarization: Component of d perpendicular to plane of incidence
Solution of wave equation for two-beam-case:
Angle between and remains constant in the investigated angular range
gk vv
0kvθB : Bragg angle
: Direction vectors of the wave vectors jkv
jsv
0dk
KkPd
0Pddk
Kk
g02g
22g
0g
gg0020
220
=⋅
χ−
−+χ−
=χ−⋅
χ−
−−
vv
vv
(5.28)
(5.29)
−
π
σ
θ= für
2cos
1P
BWith Polarization (5.30)
(5.26)Two
Beams
Dynamical Diffraction Theory
Solution of Wave Equation: Two-Beam-Case
gv
−χgvχ
0dk
KkPd
0Pddk
Kk
g02g
22g
0g
gg0020
220
=⋅
χ−
−+χ−
=χ−⋅
χ−
−−
vv
vv
• Homogenous linear system of equations• It contains the terms and • Bragg reflected wave can be reflected back into the transmitted wave• Incoming wave supplies diffracted wave and vice versa (� Pendellösung)
(5.29)
0Pk
Kk
k
Kk 2gg02
g
22g
020
220 =χχ−
χ−
−⋅
χ−
−−vv
This (secular) equation describes the shape of the dispersion surface
(5.30)
Nontrivial solution of the system of equations: det (..) = 0!
Dynamical Diffraction Theory
Solution of Wave Equation: Two-Beam-Case
0ii 1Kk: χ+−=ξ (i = 0,g) (5.31)
Distance of the origins of k0 and kg from the circles with radiusaround the reciprocal lattice points 0 and g, respectively
00 1Kk χ+=
00 2111 χ+≈χ+ (5.32) Further approximations:
22g
20 Kkk ≈≈ (5.33)
For the denominator of (5.30)
gg22
g0 PK41
vv−χχ=ξξ (5.34)
Hyperbola Equation around Lorentz point M
Dynamical Diffraction Theory
Solution of Wave Equation: Two-Beam-Case
• Shift of Laue point L towards Lorentz point M (caused by refraction)
• Removement of degeneration of point M (Breaking up of dispersion surface into two branches)
• σ and π-Polarization have to be treated independently (� four branches)
Dynamical Diffraction Theory
Solution of Wave Equation: Two-Beam-Case
Laue Point
Lorentz Point
Ewald sphere inside crystal
Ewald sphere in vacuum
• Two branches with slightly different vertical component of the wave vector � leads to beating
Dispersion Surface
gg22
g0 PK41
vv−χχ=ξξ
• Relative Splitting depends on strength of interaction• Strength of interactions depends on kind of “radiation” (X-rays, electrons, neutrons)• X-rays, 1Å, Silicon � Splitting ≈
gvχ
10-6
Dynamical Diffraction Theory
Darwin Width
Consequences: Symmetric (111)-Reflection,
270 µm thick Silicon Crystal
• finite width of Bragg reflection (Darwin width)
• pronounced Pendellösung oscillations, in particular for Laue case• shift of Bragg peak (for Bragg case) relative to vacuum Bragg angle
B0
gggD sin
1θγ
γ⋅χχ=θ∆ −
v
vv
00
gg
cos
cos
ψ=γ
ψ=γ vv
(5.35)
Dynamical Diffraction Theory
B0
gggD sin
1θγ
γ⋅χχ=θ∆ −
v
vv
Darwin width depends on:
( )( )Bg
Bg
0
g
cos
cos
θ+ψ
θ−ψ=
γ
γ=γ
v
• the Fourier components χg und χ-g
� Strength of interaction� Kind of radiation (X-rays, electrons, neutrons)� Kind of Bragg reflection (h k l)
• Wave length of X-rays
• Typical values for X-rays: E = 10 keV, Silicon, ∆θD ≅ 1 .. 5 arcsec
• Scattering geometryAsymmetry parameter
Dynamical Diffraction Theory
Darwin Width
RemarkBragg reflection g can be also reached via inversion of beam directions. There are however differences in the Darwin widths:
• Steep incidence and glancing exit:� reduced angular acceptance by the factor |γ|1/2
� enlarged divergence at the exit side by the factor |γ|-1/2
• Glancing incidence and steep exit: � enlarged angular acceptance by the factor |γ|-1/2
� reduced divergence at the exit side by the factor |γ|1/2
( )( )Bg
Bg
0
g
cos
cos
θ+ψ
θ−ψ=
γ
γ=γ
v
γ ≡ “asymmetry” parameter
B0
gggD sin
1θγ
γ⋅χχ=θ∆ −
v
vv
Dynamical Diffraction Theory
Darwin Width
These conditions play a central role in designing a high-resolution X-ray experiment
“Switching on” Absorption“ � Darwin curve becomes asymmetrical
Reason:• The two wave fields exhibit maximum intensity between and on the interatomic planes,
respectively. • When rocking the crystal the relative intensity between both is changed. • At the right side of the Darwin curve field two (here absorption is stronger) is dominant
Darwin Curve
Dynamical Diffraction Theory
Dynamische Beugungstheorie Dynamische Beugungstheorie
gg
g0
P vv
v
−χχ
γγλ=Λ
Strong reflectivity (close to one) when Bragg condition is fulfilled
Strongly reduced penetration depth of the x-rays
“Extinction”
Si 440 Reflection
λ = 1.54 A
ω = 18°
Extinction length
Extinction
Pen
etra
tion
Dep
th (µ
m)
gg
g0
P vv
v
−χχ
γγλ=Λ
Extinction Length
χ
λ∝Λ 2λ∝χ
∑+−
=j j
22j
j
0
2e
i
g
m
e
γωωωε
ρωχ
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Dielectric Polarizability
Dynamical Diffraction Theory
Extinction Length
Dynamical Theory of X-Ray Diffraction
• Dispersion surfaces describe surfaces of identical energy in k-space.
• Splitting of Dispersion surfaces at Lorentz Point � Expansion of two partial waves whichdiffer in the normal component of the wave vectors.
Calculation of the electronic structure in solid materials
• Completely analogue treatment
• Result of theory: Function E(k)
• Energy band gap at the border of the Brillouin Zone depends on the strength of interaction between the electrons with the solid material.
• At the border of the Brillouin-Zone (i.e. k = ± π/a) Bragg reflection of the electrons take place.
• We can – alternatively - state that, Bragg reflection at the border of the Brillouin zone leads to the appearance of a band gap.
Analogy to Electron Band Structure
Dynamical Diffraction Theory
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